\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  27.0 & -7.00 x_{1} & -5.00 x_{2} & +  2.00 x_{3} &   & -6.00 x_{5} & +  9.00 x_{6} & -10.00 x_{7} & -10.00 x_{8} & +  7.00 x_{9}\\
 x_{11}   &  -3.0 & -6.00 x_{1} & -1.00 x_{2} & -2.00 x_{3} & +  2.00 x_{4} & +  8.00 x_{5} & +  8.00 x_{6} & -9.00 x_{7} & -8.00 x_{8} & +  4.00 x_{9}\\
 x_{12}   &  18.0 & -5.00 x_{1} & + 10.00 x_{2} &   & -6.00 x_{4} & -5.00 x_{5} & +  8.00 x_{6} & -4.00 x_{7} & -6.00 x_{8} & +  5.00 x_{9}\\
 x_{13}   &  -25.0  &   & -9.00 x_{2} & -1.00 x_{3} & +  7.00 x_{4} &   & -2.00 x_{6} & +  4.00 x_{7} & -6.00 x_{8} & +  9.00 x_{9}\\
 x_{14}   &  28.0 & -5.00 x_{1} & +  9.00 x_{2} & +  9.00 x_{3} & -9.00 x_{4} & -5.00 x_{5} & -1.00 x_{6} & +  3.00 x_{7} &   & -9.00 x_{9}\\
 x_{15}   &  35.0 & +  1.00 x_{1} & +  5.00 x_{2} & +  8.00 x_{3} & -1.00 x_{4} & -9.00 x_{5} & -6.00 x_{6} & -9.00 x_{7} & +  5.00 x_{8} & +  4.00 x_{9}\\
\hline
z    &  0.0 & -1.00 x_{1} & -4.00 x_{2} & +  2.00 x_{3} & +  3.00 x_{4} & +  4.00 x_{5} & -5.00 x_{6} &   & +  5.00 x_{8} & -4.00 x_{9}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{9}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,15$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  7.00 y_{10} & +  6.00 y_{11} & +  5.00 y_{12} &   & +  5.00 y_{14} & -1.00 y_{15}\\
 y_{2}   &  1.0 & +  5.00 y_{10} & +  1.00 y_{11} & -10.00 y_{12} & +  9.00 y_{13} & -9.00 y_{14} & -5.00 y_{15}\\
 y_{3}   &  1.0 & -2.00 y_{10} & +  2.00 y_{11} &   & +  1.00 y_{13} & -9.00 y_{14} & -8.00 y_{15}\\
 y_{4}   &  1.0  &   & -2.00 y_{11} & +  6.00 y_{12} & -7.00 y_{13} & +  9.00 y_{14} & +  1.00 y_{15}\\
 y_{5}   &  1.0 & +  6.00 y_{10} & -8.00 y_{11} & +  5.00 y_{12} &   & +  5.00 y_{14} & +  9.00 y_{15}\\
 y_{6}   &  1.0 & -9.00 y_{10} & -8.00 y_{11} & -8.00 y_{12} & +  2.00 y_{13} & +  1.00 y_{14} & +  6.00 y_{15}\\
 y_{7}   &  1.0 & + 10.00 y_{10} & +  9.00 y_{11} & +  4.00 y_{12} & -4.00 y_{13} & -3.00 y_{14} & +  9.00 y_{15}\\
 y_{8}   &  1.0 & + 10.00 y_{10} & +  8.00 y_{11} & +  6.00 y_{12} & +  6.00 y_{13} &   & -5.00 y_{15}\\
 y_{9}   &  1.0 & -7.00 y_{10} & -4.00 y_{11} & -5.00 y_{12} & -9.00 y_{13} & +  9.00 y_{14} & -4.00 y_{15}\\
\hline
z    &  -0 & -27.00 y_{10} & +  3.00 y_{11} & -18.00 y_{12} & + 25.00 y_{13} & -28.00 y_{14} & -35.00 y_{15}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 4 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  7.00 y_{10} & +  6.00 y_{11} & +  5.00 y_{12} &   & +  5.00 y_{14} & -1.00 y_{15}\\
 y_{2}   &  2.0 & -2.00 y_{10} & -3.00 y_{11} & -15.00 y_{12} & -1.00 y_{9} &   & -9.00 y_{15}\\
 y_{3}   &  1.11111111111 & -2.78 y_{10} & +  1.56 y_{11} & -0.56 y_{12} & -0.11 y_{9} & -8.00 y_{14} & -8.44 y_{15}\\
 y_{4}   &  0.222222222222 & +  5.44 y_{10} & +  1.11 y_{11} & +  9.89 y_{12} & +  0.78 y_{9} & +  2.00 y_{14} & +  4.11 y_{15}\\
 y_{5}   &  1.0 & +  6.00 y_{10} & -8.00 y_{11} & +  5.00 y_{12} &   & +  5.00 y_{14} & +  9.00 y_{15}\\
 y_{6}   &  1.22222222222 & -10.56 y_{10} & -8.89 y_{11} & -9.11 y_{12} & -0.22 y_{9} & +  3.00 y_{14} & +  5.11 y_{15}\\
 y_{7}   &  0.555555555556 & + 13.11 y_{10} & + 10.78 y_{11} & +  6.22 y_{12} & +  0.44 y_{9} & -7.00 y_{14} & + 10.78 y_{15}\\
 y_{8}   &  1.66666666667 & +  5.33 y_{10} & +  5.33 y_{11} & +  2.67 y_{12} & -0.67 y_{9} & +  6.00 y_{14} & -7.67 y_{15}\\
 y_{13}   &  0.111111111111 & -0.78 y_{10} & -0.44 y_{11} & -0.56 y_{12} & -0.11 y_{9} & +  1.00 y_{14} & -0.44 y_{15}\\
\hline
z    &  2.77777777778 & -46.44 y_{10} & -8.11 y_{11} & -31.89 y_{12} & -2.78 y_{9} & -3.00 y_{14} & -46.11 y_{15}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  46.4444444444 & -7.00 x_{1} & +  2.00 x_{2} & +  2.78 x_{3} & -5.44 x_{4} & -6.00 x_{5} & + 10.56 x_{6} & -13.11 x_{7} & -5.33 x_{8} & +  0.78 x_{13}\\
 x_{11}   &  8.11111111111 & -6.00 x_{1} & +  3.00 x_{2} & -1.56 x_{3} & -1.11 x_{4} & +  8.00 x_{5} & +  8.89 x_{6} & -10.78 x_{7} & -5.33 x_{8} & +  0.44 x_{13}\\
 x_{12}   &  31.8888888889 & -5.00 x_{1} & + 15.00 x_{2} & +  0.56 x_{3} & -9.89 x_{4} & -5.00 x_{5} & +  9.11 x_{6} & -6.22 x_{7} & -2.67 x_{8} & +  0.56 x_{13}\\
 x_{9}   &  2.77777777778  &   & +  1.00 x_{2} & +  0.11 x_{3} & -0.78 x_{4} &   & +  0.22 x_{6} & -0.44 x_{7} & +  0.67 x_{8} & +  0.11 x_{13}\\
 x_{14}   &  3.0 & -5.00 x_{1} &   & +  8.00 x_{3} & -2.00 x_{4} & -5.00 x_{5} & -3.00 x_{6} & +  7.00 x_{7} & -6.00 x_{8} & -1.00 x_{13}\\
 x_{15}   &  46.1111111111 & +  1.00 x_{1} & +  9.00 x_{2} & +  8.44 x_{3} & -4.11 x_{4} & -9.00 x_{5} & -5.11 x_{6} & -10.78 x_{7} & +  7.67 x_{8} & +  0.44 x_{13}\\
\hline
z    &  -2.77777777778 & -1.00 x_{1} & -2.00 x_{2} & -1.11 x_{3} & -0.22 x_{4} & -1.00 x_{5} & -1.22 x_{6} & -0.56 x_{7} & -1.67 x_{8} & -0.11 x_{13}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  46.4444444444 & -7.00 x_{1} & +  2.00 x_{2} & +  2.78 x_{3} & -5.44 x_{4} & -6.00 x_{5} & + 10.56 x_{6} & -13.11 x_{7} & -5.33 x_{8} & +  0.78 x_{13}\\
 x_{11}   &  8.11111111111 & -6.00 x_{1} & +  3.00 x_{2} & -1.56 x_{3} & -1.11 x_{4} & +  8.00 x_{5} & +  8.89 x_{6} & -10.78 x_{7} & -5.33 x_{8} & +  0.44 x_{13}\\
 x_{12}   &  31.8888888889 & -5.00 x_{1} & + 15.00 x_{2} & +  0.56 x_{3} & -9.89 x_{4} & -5.00 x_{5} & +  9.11 x_{6} & -6.22 x_{7} & -2.67 x_{8} & +  0.56 x_{13}\\
 x_{9}   &  2.77777777778  &   & +  1.00 x_{2} & +  0.11 x_{3} & -0.78 x_{4} &   & +  0.22 x_{6} & -0.44 x_{7} & +  0.67 x_{8} & +  0.11 x_{13}\\
 x_{14}   &  3.0 & -5.00 x_{1} &   & +  8.00 x_{3} & -2.00 x_{4} & -5.00 x_{5} & -3.00 x_{6} & +  7.00 x_{7} & -6.00 x_{8} & -1.00 x_{13}\\
 x_{15}   &  46.1111111111 & +  1.00 x_{1} & +  9.00 x_{2} & +  8.44 x_{3} & -4.11 x_{4} & -9.00 x_{5} & -5.11 x_{6} & -10.78 x_{7} & +  7.67 x_{8} & +  0.44 x_{13}\\
\hline
z    &  -11.1111111111 & -1.00 x_{1} & -8.00 x_{2} & +  1.56 x_{3} & +  6.11 x_{4} & +  4.00 x_{5} & -5.89 x_{6} & +  1.78 x_{7} & +  2.33 x_{8} & -0.44 x_{13}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  46.4444444444 & -7.00 x_{1} & +  2.00 x_{2} & +  2.78 x_{3} & -5.44 x_{4} & -6.00 x_{5} & + 10.56 x_{6} & -13.11 x_{7} & -5.33 x_{8} & +  0.78 x_{13}\\
 x_{11}   &  8.11111111111 & -6.00 x_{1} & +  3.00 x_{2} & -1.56 x_{3} & -1.11 x_{4} & +  8.00 x_{5} & +  8.89 x_{6} & -10.78 x_{7} & -5.33 x_{8} & +  0.44 x_{13}\\
 x_{12}   &  31.8888888889 & -5.00 x_{1} & + 15.00 x_{2} & +  0.56 x_{3} & -9.89 x_{4} & -5.00 x_{5} & +  9.11 x_{6} & -6.22 x_{7} & -2.67 x_{8} & +  0.56 x_{13}\\
 x_{9}   &  2.77777777778  &   & +  1.00 x_{2} & +  0.11 x_{3} & -0.78 x_{4} &   & +  0.22 x_{6} & -0.44 x_{7} & +  0.67 x_{8} & +  0.11 x_{13}\\
 x_{14}   &  3.0 & -5.00 x_{1} &   & +  8.00 x_{3} & -2.00 x_{4} & -5.00 x_{5} & -3.00 x_{6} & +  7.00 x_{7} & -6.00 x_{8} & -1.00 x_{13}\\
 x_{15}   &  46.1111111111 & +  1.00 x_{1} & +  9.00 x_{2} & +  8.44 x_{3} & -4.11 x_{4} & -9.00 x_{5} & -5.11 x_{6} & -10.78 x_{7} & +  7.67 x_{8} & +  0.44 x_{13}\\
\hline
z    &  -11.1111111111 & -1.00 x_{1} & -8.00 x_{2} & +  1.56 x_{3} & +  6.11 x_{4} & +  4.00 x_{5} & -5.89 x_{6} & +  1.78 x_{7} & +  2.33 x_{8} & -0.44 x_{13}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  60.9285714286 & -17.71 x_{1} & +  7.36 x_{2} & -1.79 x_{11} & -7.43 x_{4} & +  8.29 x_{5} & + 26.43 x_{6} & -32.36 x_{7} & -14.86 x_{8} & +  1.57 x_{13}\\
 x_{3}   &  5.21428571429 & -3.86 x_{1} & +  1.93 x_{2} & -0.64 x_{11} & -0.71 x_{4} & +  5.14 x_{5} & +  5.71 x_{6} & -6.93 x_{7} & -3.43 x_{8} & +  0.29 x_{13}\\
 x_{12}   &  34.7857142857 & -7.14 x_{1} & + 16.07 x_{2} & -0.36 x_{11} & -10.29 x_{4} & -2.14 x_{5} & + 12.29 x_{6} & -10.07 x_{7} & -4.57 x_{8} & +  0.71 x_{13}\\
 x_{9}   &  3.35714285714 & -0.43 x_{1} & +  1.21 x_{2} & -0.07 x_{11} & -0.86 x_{4} & +  0.57 x_{5} & +  0.86 x_{6} & -1.21 x_{7} & +  0.29 x_{8} & +  0.14 x_{13}\\
 x_{14}   &  44.7142857143 & -35.86 x_{1} & + 15.43 x_{2} & -5.14 x_{11} & -7.71 x_{4} & + 36.14 x_{5} & + 42.71 x_{6} & -48.43 x_{7} & -33.43 x_{8} & +  1.29 x_{13}\\
 x_{15}   &  90.1428571429 & -31.57 x_{1} & + 25.29 x_{2} & -5.43 x_{11} & -10.14 x_{4} & + 34.43 x_{5} & + 43.14 x_{6} & -69.29 x_{7} & -21.29 x_{8} & +  2.86 x_{13}\\
\hline
z    &  -3.0 & -7.00 x_{1} & -5.00 x_{2} & -1.00 x_{11} & +  5.00 x_{4} & + 12.00 x_{5} & +  3.00 x_{6} & -9.00 x_{7} & -3.00 x_{8} &   \\
\end{array}\]


 $ x_{4} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  35.8055555556 & -12.56 x_{1} & -4.25 x_{2} & -1.53 x_{11} & +  0.72 x_{12} & +  9.83 x_{5} & + 17.56 x_{6} & -25.08 x_{7} & -11.56 x_{8} & +  1.06 x_{13}\\
 x_{3}   &  2.79861111111 & -3.36 x_{1} & +  0.81 x_{2} & -0.62 x_{11} & +  0.07 x_{12} & +  5.29 x_{5} & +  4.86 x_{6} & -6.23 x_{7} & -3.11 x_{8} & +  0.24 x_{13}\\
 x_{4}   &  3.38194444444 & -0.69 x_{1} & +  1.56 x_{2} & -0.03 x_{11} & -0.10 x_{12} & -0.21 x_{5} & +  1.19 x_{6} & -0.98 x_{7} & -0.44 x_{8} & +  0.07 x_{13}\\
 x_{9}   &  0.458333333333 & +  0.17 x_{1} & -0.12 x_{2} & -0.04 x_{11} & +  0.08 x_{12} & +  0.75 x_{5} & -0.17 x_{6} & -0.38 x_{7} & +  0.67 x_{8} & +  0.08 x_{13}\\
 x_{14}   &  18.625 & -30.50 x_{1} & +  3.38 x_{2} & -4.88 x_{11} & +  0.75 x_{12} & + 37.75 x_{5} & + 33.50 x_{6} & -40.88 x_{7} & -30.00 x_{8} & +  0.75 x_{13}\\
 x_{15}   &  55.8402777778 & -24.53 x_{1} & +  9.44 x_{2} & -5.08 x_{11} & +  0.99 x_{12} & + 36.54 x_{5} & + 31.03 x_{6} & -59.35 x_{7} & -16.78 x_{8} & +  2.15 x_{13}\\
\hline
z    &  13.9097222222 & -10.47 x_{1} & +  2.81 x_{2} & -1.17 x_{11} & -0.49 x_{12} & + 10.96 x_{5} & +  8.97 x_{6} & -13.90 x_{7} & -5.22 x_{8} & +  0.35 x_{13}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  20.2222222222 & -18.22 x_{1} & + 34.00 x_{9} & -0.11 x_{11} & -2.11 x_{12} & -15.67 x_{5} & + 23.22 x_{6} & -12.33 x_{7} & -34.22 x_{8} & -1.78 x_{13}\\
 x_{3}   &  5.77777777778 & -2.28 x_{1} & -6.50 x_{9} & -0.89 x_{11} & +  0.61 x_{12} & + 10.17 x_{5} & +  3.78 x_{6} & -8.67 x_{7} & +  1.22 x_{8} & +  0.78 x_{13}\\
 x_{4}   &  9.11111111111 & +  1.39 x_{1} & -12.50 x_{9} & -0.56 x_{11} & +  0.94 x_{12} & +  9.17 x_{5} & -0.89 x_{6} & -5.67 x_{7} & +  7.89 x_{8} & +  1.11 x_{13}\\
 x_{2}   &  3.66666666667 & +  1.33 x_{1} & -8.00 x_{9} & -0.33 x_{11} & +  0.67 x_{12} & +  6.00 x_{5} & -1.33 x_{6} & -3.00 x_{7} & +  5.33 x_{8} & +  0.67 x_{13}\\
 x_{14}   &  31.0 & -26.00 x_{1} & -27.00 x_{9} & -6.00 x_{11} & +  3.00 x_{12} & + 58.00 x_{5} & + 29.00 x_{6} & -51.00 x_{7} & -12.00 x_{8} & +  3.00 x_{13}\\
 x_{15}   &  90.4444444444 & -11.94 x_{1} & -75.50 x_{9} & -8.22 x_{11} & +  7.28 x_{12} & + 93.17 x_{5} & + 18.44 x_{6} & -87.67 x_{7} & + 33.56 x_{8} & +  8.44 x_{13}\\
\hline
z    &  24.2222222222 & -6.72 x_{1} & -22.50 x_{9} & -2.11 x_{11} & +  1.39 x_{12} & + 27.83 x_{5} & +  5.22 x_{6} & -22.33 x_{7} & +  9.78 x_{8} & +  2.22 x_{13}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  1.29078014184 & -1.16 x_{1} & +  2.17 x_{9} & -0.01 x_{11} & -0.13 x_{12} & -0.06 x_{10} & +  1.48 x_{6} & -0.79 x_{7} & -2.18 x_{8} & -0.11 x_{13}\\
 x_{3}   &  18.9007092199 & -14.10 x_{1} & + 15.56 x_{9} & -0.96 x_{11} & -0.76 x_{12} & -0.65 x_{10} & + 18.85 x_{6} & -16.67 x_{7} & -20.99 x_{8} & -0.38 x_{13}\\
 x_{4}   &  20.9432624113 & -9.27 x_{1} & +  7.39 x_{9} & -0.62 x_{11} & -0.29 x_{12} & -0.59 x_{10} & + 12.70 x_{6} & -12.88 x_{7} & -12.13 x_{8} & +  0.07 x_{13}\\
 x_{2}   &  11.4113475177 & -5.65 x_{1} & +  5.02 x_{9} & -0.38 x_{11} & -0.14 x_{12} & -0.38 x_{10} & +  7.56 x_{6} & -7.72 x_{7} & -7.77 x_{8} & -0.01 x_{13}\\
 x_{14}   &  105.865248227 & -93.46 x_{1} & + 98.87 x_{9} & -6.41 x_{11} & -4.82 x_{12} & -3.70 x_{10} & + 114.97 x_{6} & -96.66 x_{7} & -138.70 x_{8} & -3.58 x_{13}\\
 x_{15}   &  210.70212766 & -120.31 x_{1} & + 126.69 x_{9} & -8.88 x_{11} & -5.28 x_{12} & -5.95 x_{10} & + 156.54 x_{6} & -161.01 x_{7} & -169.96 x_{8} & -2.13 x_{13}\\
\hline
z    &  60.1489361702 & -39.10 x_{1} & + 37.90 x_{9} & -2.31 x_{11} & -2.36 x_{12} & -1.78 x_{10} & + 46.48 x_{6} & -44.24 x_{7} & -51.02 x_{8} & -0.94 x_{13}\\
\end{array}\]


 $ x_{6} $ enters and Unbounded Dictionary!


 $ x_{6} $ enters and Unbounded Dictionary!
\end{document}
